We study the class of functions $f$ on $mathbb{R}$ satisfying a Lipschitz estimate in the Schatten ideal $mathcal{L}_p$ for $0 < p leq 1$. The corresponding problem with $pgeq 1$ has been extensively studied, but the quasi-Banach range $0 < p < 1$ is by comparison poorly understood. Using techniques from wavelet analysis, we prove that Lipschitz functions belonging to the homogeneous Besov class $dot{B}^{frac{1}{p}}_{frac{p}{1-p},p}(mathbb{R})$ obey the estimate $$ |f(A)-f(B)|_{p} leq C_{p}(|f|_{L_{infty}(mathbb{R})}+|f|_{dot{B}^{frac{1}{p}}_{frac{p}{1-p},p}(mathbb{R})})|A-B|_{p} $$ for all bounded self-adjoint operators $A$ and $B$ with $A-Bin mathcal{L}_p$. In the case $p=1$, our methods recover and provide a new perspective on a result of Peller that $f in dot{B}^1_{infty,1}$ is sufficient for a function to be Lipschitz in $mathcal{L}_1$. We also provide related Holder-type estimates, extending results of Aleksandrov and Peller. In addition, we prove the surprising fact that non-constant periodic functions on $mathbb{R}$ are not Lipschitz in $mathcal{L}_p$ for any $0 < p < 1$. This gives counterexamples to a 1991 conjecture of Peller that $f in dot{B}^{1/p}_{infty,p}(mathbb{R})$ is sufficient for $f$ to be Lipschitz in $mathcal{L}_p$.
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